Positive periodic solutions of delayed differential equations.(English)Zbl 1248.34104

Summary: In this paper, we are concerned with the existence, multiplicity and nonexistence of positive ?-periodic solutions of the following equation $u''(t)+ a(t,u) u(t)=\lambda b(t) f(u(t-\tau(t))),\quad t\in\mathbb{R},$ where $$a(\cdot,\cdot)\in C(\mathbb{R}\times\mathbb{R}, \mathbb{R}^+)$$ is a $$\omega$$-periodc function with respect to the first variable, $$b(\cdot)\in C(\mathbb{R},[0,\infty))$$, $$\tau(\cdot)\in C(\mathbb{R}, \mathbb{R})$$ are $$\omega$$-periodic functions, $$f\in C([0,\infty), [0,\infty))$$ and $$f(s)> 0$$ for $$s> 0$$, $$\lambda> 0$$ is a parameter. The proof of our main result is based upon fixed point index theory.

MSC:

 34K13 Periodic solutions to functional-differential equations 47N20 Applications of operator theory to differential and integral equations
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References:

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